Isosurface Extraction and Spatial Filtering Using Persistent Octree (POT)

1
Isosurface Extraction and Spatial Filtering Using
Persistent Octree (POT)

• Qingmin Shi and Joseph JaJa
• Institute for Advanced Computer Studies and
• Department of Electrical and Computer Engineering
• University of Maryland, College Park

2
Volumetric Data Visualization
NLM Visible Human (Lorensen Vis95)
LLNL Richtmyer-Meshkov Instability (Shi, JaJa
Vis06)
Tetra Meshes of Ribosome 30S (Zhang, Xu, and
Bajaj, Computer Aided Geometric Design 2006)
3
Isosurface Extraction and Rendering

• What is isosurface?

Isosurface A 3D-contour -- A 2D-surface embedded
in a 3D space with constant density (isovalue) C.
Image created from the UNC CT Head data set
using VolPack Lacroute and Levoy, SIGGRAPH 94.
4
Basic Steps in Isosurface Computation

• Finding the active cells that are cut by the
  isosurface.
• Piece-wise linear approximation (triangulation)
  of the isosurface patches in each active cell.
• The marching cubes algorithm was used in this
  paper.
• Rendering triangles using either hardware or
  software.

5
Identifying Active Cells

• Why is this step important?
• Volume size can be very large.
• The size of the whole Richtmyer-Meshkov
  instability data set is about 2.1 TB!
• The isosurface in general only occupies a small
  subspace.
• Examining all cells is very expensive and
  unnecessary.
• Solution value based partitioning.

6
Active Cell Identification As A Stabbing Query

• Let mini and maxi be the minimum and maximum
  scalar value of all the vertices of a cell i.
• A cell is active if and only if the isovalue C is
  within the range mini, maxi.
• Finding active cells is equivalent to stabbing a
  set of horizontal segments with an infinite
  vertical line.

C
7
Solutions to the Stabbing Query

• Optimal solutions
• Priority tree (McCreight SIAM J. Comp.85).
• Interval tree (Cignoni et al. Volvis96, Chiang
  and Silva Vis97).
• Space O(N), query time O(log N K)
• N number of cells K number of active cells.
• Good practical solutions
• Span space (Livnat, Shen, Johnson Vis96).
• Span triangles (von Rymon-Lipinski et al.
  Vis04).
• Fixed-sized buckets (Waters, Co, and Joy
  EuroVis05).

8
Spatial Filtering

• Only a subset of the active cells need to be
  processed
• View-dependent rendering cells that are
  visible.
• Ray casting the first active cell that
  intersects each ray.
• 4D isosurface visualization cells that
  intersect the cutting hyperplane.
• Benefits
• Reduced indexing structure search time.
• Reduced triangulation time.
• Reduced rendering time.

9
View-dependent Isosurface Rendering

• Render the isosurface patches in a front-to-back
  order.
• Keep a coverage mask to tell which part of the
  screen has been covered.
• Do not triangulate and render active cells whose
  projection to the screen has already been covered.

Problem How to examine the active cells in a
front-to-back order?
10
Value-based vs. Location-based Partitioning

• Previous view-dependent algorithms often employ
  location-based partitioning.
• Example a min-max octree equipped with min, max
  values for each internal node.
• Problem identifying active cells is inefficient.
• Active cells reported by a value-based
  partitioning structure do not obey any spatial
  ordering.

11
Our Solution

• Build one octree to organize the active cells
  corresponding to EACH possible isovalue.
• Compress the sequence of octrees into a
  persistent octree (POT).
• The benefits of both approaches
• The size of the POT is linear in the input size
  O(N), even for continuous data types!
• Identifying visible active cells is done by a
  front-to-back traversal of an octree that
  consists of ONLY the active cells O(K)!
• Spatial filtering is efficient using the octree.

12
Persistent Data Structures

• Consider a dynamic data structure D.
• Each insertion/deletion produces a new version of
  D.
• The entire evolution history of D is recorded as
  P.
• Any particular version of D can be searched by
  knowing its version number.
• If the cost of each update is small, the entire
  persistent data structure is small.

D(0)
D(2)
D(3)
D(1)
A persistent data structure P
13
Handling Stabbing Queries Using Persistent Data
Structures
S1
S2
S3
S4
S5
scalar value
V1
V0
V5
V7
V6
V9
V8
V10
V3
V2
V4
14
A Standard Octree

• Orange nodes represent active cells (regions).
• Gray nodes represent mixed regions.
• White (non-active) nodes can be omitted.
• Not suitable to be made persistent.
• Why? A single update operation requires
  significant change of the data structure.

15
The Compact Octree

• Whats good about the compact octree?
• On average, only a CONSTANT number of changes
  need to be made to the tree.
• So?
• The corresponding persistent octree requires
  LINEAR space.

16
Update the Compact Octree The Insertion
Case 2
Case 1
Case 3
17
Update the Compact Octree The Deletion

• An expensive delete operation.
• Fortunately, it does not happen very often.
• Amortized update complexity still O(1).
• Not true for arbitrary delete/insert operation.

18
The Persistent Octree An 1-D Illustration
19
System Setup

• A Linux PC with dual Xeon 3.0 GHz processors
  (only one was used).
• 8 GB memory.
• 150 GB local disk.
• NVidia6800 Ultra GPU card.

20
Data Sets

• Downsampled LLNL Richtmyer-Meshkov Instability
  Data Set (byte, 1024 x 1024 x 960 x 35 time
  steps).
• Stanford Bunny (short, 512 x 512 x 360).
• UNC MR Brain (short, 256 x 256 x 109).
• Head Aneurysm (byte 512 x 512 x 512) (Michael
  Meißner, Viatronix Inc., USA).

21
Storage Cost
Data Set Data Set Richtmyer-Meshkov Instability UNC MR Brain Head Aneurysm Standford Bunny
Storage Cost (byte) BONO 144M 1.5M 19M 20M
Storage Cost (byte) POT 230M 5.0M 76M 62M

• The POT is about 1.6 to 4 times bigger.
• Both BONO and POT are built on 4x4x4 cell
  blocks.
• The above table only lists the sizes of the
  indexing structures. The sizes of the
• raw data sets are the same for both BONO and
  POT and thus are omitted here.
• Wilhelms and Van Gelder, ACM Trans. on
  Graphics, 92.

22
Performance Comparison of View-independent
Isosurface Extraction
Data Set Data Set Richtmyer-Meshkov Instability Stanford Bunny UNC MR Brain Head Aneurysm
Isovalue Isovalue 190 1750 1750 55
BONO Index search time (ms) 710 64 64 22
POT Index search time (ms) 401 61 61 15
Speedup of POT index searching time relative to BONO Speedup of POT index searching time relative to BONO 1.77 1.05 1.21 1.47
23
Performance Comparison of View-dependent
Isosurface Extraction
Data Set Data Set Richtmyer-Mashkov Instability Stanford Bunny UNC MR Brain Head Aneurysm
BONO of nodes visited 212,992 19,897 7,318 17,260
BONO Execution time (ms) 10,877 1,130 614 938
POT of nodes visited 142,157 6,076 5,368 6,467
POT Execution time (ms) 9,511 1,040 602 806
Speedup of POT relative to BONO of nodes visited 1.50 3.27 1.36 2.67
Speedup of POT relative to BONO Execution time 1.14 1.09 1.02 1.16
24
Other Applications of POT

• Isosurface rendering by Ray-casting.
• Go straight to the intersecting active cell.
• Rendering time-varying data (4-D isosurface)
• A 4-D isosurface can only be visualized by
  slicing it using a 4-D hyperplane.
• POT can be used to quickly identify cells that
  are both active and relevant (cut by the
  hyperplane).
• Multi-resolution isosurface rendering
• Triangulate supercells at different resolutions.
• Internal nodes in a POT naturally represent
  active supercells.

25
Conclusions

• Good things about POT.
• Simultaneous pruning in both value and viewing
  space.
• Fast search time and reasonable storage cost.
• Hierarchical organization of active cells
  facilitates spatial search.
• Further improvements.
• Need tests on continuous data types.
• Improve the search time and storage cost.
• How to incorporate this technique into
  parallel/distributed isosurface rendering systems.

26
Thank You!

• Questions?

27
Isosurfacing By Ray Casting

• The basic scheme
• Shoot a ray from each pixel towards the volume.
• Identify the foremost active cell intersected by
  the ray.
• Analytically compute the position of the
  intersection point.
• Shade the intersection point.
• The bottle neck traversal of non-active cells.
• Using POT
• No active cells in the octree.
• No need to examine non-active regions.
• An example
• Standard ray-casting 9 cells to examine.
• POT 3 nodes to visit.

28
An Example of VD Isosurfacing
A front view of the MRI Head
The side view
The final coverage mask
29
More Pictures
A top view of the MRI-brain data
Stanford bunny
Cutting plane X500
Cutting plane Z650
30
Practical Issues

• Build POT on 4x4x4 cubes rather than individual
  cells.
• Sequentialize the POT (an acyclic graph) on disk.
• Use hierarchical coverage mask to speedup
  visibility test (Greene SIGGRAPH96, Livnat and
  Hansen Vis98).